Asymptotic Analysis 54 (2007) 103–123
IOS Press
103
Error estimates for periodic homogenization
with non-smooth coefficients
D. Onofrei and B. Vernescu
Department of Mathematical Sciences, Worcester Polytechnic Institute, 100 Institute Rd., Worcester,
MA 01609, USA
Abstract. In this paper we present new results regarding the H01 -norm error estimate for the classical problem in homogenization using suitable boundary layer correctors. Compared with all the existing results on the subject, which assume either smooth
enough coefficients or smooth data, we use the periodic unfolding method and propose a new asymptotic series to approximate
the solution uε with an error estimate which holds true for nonsmooth coefficients and general data.
1. Introduction
This paper is dedicated to the study of the error estimates for the classical problem in homogenization
using suitable boundary layer correctors.
Let Ω ∈ RN , denote a bounded convex polyhedron or a convex bounded domain with a sufficiently
smooth boundary. Consider also the unit cube Y = (0, 1)N . It is well known that for A ∈ L∞ (Y )N ×N ,
Y -periodic with m|ξ|2 Aij (y)ξi ξj M |ξ|2 , ∀ξ ∈ RN the solutions of


x
−∇ · A
∇uε (x) = f
ε

uε = 0
in Ω,
(1.1)
on ∂Ω
have the property that (see [18,12,3,4]),
uε u0
in H01 (Ω),
where u0 verifies
−∇ · Ahom ∇u0 (x) = f
u0 = 0
in Ω,
on ∂Ω
∂χ
j
1
with Ahom
ij = MY (Aij (y)+Aik (y) ∂yk ), MY (·) = |Y |
MY (χ) = 0} are the solutions of the local problem
(1.2)
Y
1
· dy and where χj ∈ Wper (Y ) = {χ ∈ Hper
(Y ) |
−∇y · A(y)(∇χj + ej ) = 0
0921-7134/07/$17.00  2007 – IOS Press and the authors. All rights reserved
(1.3)
104
D. Onofrei and B. Vernescu / Error estimates for periodic homogenization with non-smooth coefficients
and where ej denote the vectors of the canonical basis in RN .
We mention that, throughout this paper, ∇ and (∇·) will denote the full gradient and divergence
operators respectively, and with ∇x , (∇x ·) and ∇y , (∇y ·) we will denote the gradient and the divergence
in the slow and fast variable respectively.
We will also denote, throughout the paper, by Φ the continuous extension of any arbitrary function
Φ ∈ W p,m (Ω) with p, m ∈ Z, to the space W p,m (RN ). With minimal assumption on the smoothness of
Ω a stable extension operator can be constructed (see [20], Ch. VI, 3.1).
The formal asymptotic expansion corresponding to the above results can be written as
uε (x) = u0 (x) + εw1
x
x,
+ ···,
ε
where
w1
x
x,
ε
= χj
x ∂u0
.
ε ∂xj
(1.4)
We make the observation that the summation convention over repeated indices will be used in the
remaining of the chapter and that the letter C will denote a constant independent of any other parameter,
otherwise specified.
A classical result (see [18,12,15,3]), states that with additional regularity assumption on χj , the solutions of the local problems, one has
uε (·) − u0 (·) − εw1 ·, . ε 1
H 1 (Ω )
Cε 2 .
(1.5)
Without any additional assumptions a similar result has been recently proved by G. Griso in [10],
using the Periodic Unfolding method developed in [7], i.e.,
uε (·) − u0 (·) − εχj . Qε ∂u0 ε
∂x j
1
H 1 (Ω )
Cε 2 u0 H 2 (Ω )
(1.6)
with
x ∈ Ω̃ε ,
Qε (φ)(x) =
MYε (φ)(εξ
+
εi)x̄i1,1ξ
· ··· ·
i1 ,...,iN
x̄iNN,ξ ,
ξ=
x
ε
for φ ∈ L2 (Ω), i = (i1 , . . . , iN ) ∈ {0, 1}N and
x̄ikk,ξ

xk − εξk



ε
=


 1 − xk − εξk
ε
where MYε (φ)(x) =
1
εN
εξ+εY
if ik = 1,
x ∈ ε(ξ + Y ),
if ik = 0,
φ(y) dy and Ω̃ε =
ξ {εξ
+ εY , with (εξ + εY ) ∩ Ω = ∅}.
D. Onofrei and B. Vernescu / Error estimates for periodic homogenization with non-smooth coefficients
105
In order to improve the error estimates in (1.5) boundary layer terms have been introduced as solutions
to
−∇ · A
x
∇θε = 0 in Ω,
ε
θε = w1 x,
x
ε
on ∂Ω.
(1.7)
With the assumptions that A ∈ C ∞ (Y ) and is a Y -periodic matrix and that the homogenized solution
u0 is sufficiently smooth, it has been proved in [4] (see also [15]) that
uε (·) − u0 (·) − εw1 ·, . + εθε (·)
ε
Cε,
(1.8)
uε (·) − u0 (·) − εw1 ·, . + εθε (·)
ε
Cε2 .
(1.9)
H01 (Ω )
L2 (Ω )
Moskow and Vogelius [16], proved the above estimates assuming A ∈ C ∞ (Y ), Y -periodic matrix
and u0 ∈ H 2 (Ω) or u0 ∈ H 3 (Ω) for (1.8) or (1.9) respectively. Inequality (1.8) is proved in Allaire and
Amar [1] for the case when A ∈ L∞ (Y ) and u0 ∈ W 2,∞ (Ω). J. Casado-Diaz proved in [5] that (1.8)
holds true for Ω ∈ C 1,1 and f ∈ LN +τ (Ω) for some τ > 0.
In [21], Sarkis and Versieux showed that estimates (1.8) and (1.9) respectively still hold in a more
1,q
1,q
(Y ) for (1.8), and u0 ∈ W 3,p (Ω), χj ∈ Wper
(Y )
general setting, when one has u0 ∈ W 2,p (Ω), χj ∈ Wper
1
1
1
for (1.9), where, in both cases, p > N and q > N satisfy p + q 2 . In [21] the constants in the right
hand side of (1.8) and (1.9) are proportional to u0 W 2,p (Ω ) and u0 W 3,p (Ω ) respectively.
In order to improve the error estimate in (1.9) one needs to consider the second order boundary layer
corrector ϕε defined as the solution of,
−∇ · A
x
∇ϕε = 0
ε
in Ω,
ϕε (x) = χij
x ∂2 u 0
ε ∂xi ∂xj
on ∂Ω,
(1.10)
where χij ∈ Wper (Y ) are solution of the following local problems,
∇y · (A∇y χij ) = bij + Ahom
ij
(1.11)
∂χ
j
∂
with Ahom defined by (1.2), MY (bij (y)) = −Ahom
ij , and bij = −Aij − Aik ∂yk − ∂yk (Aik χj ).
In this paper, we answer the open question about error estimates for (1.1) in general convex domains
and coefficients A ∈ L∞ (Y ). Inspired by an idea of Griso presented in [10], we use the periodic unfolding method developed by Cioranescu, Damlamian and Griso [7] and a general smoothing argument to
replace in (1.8) and (1.9) w1 (x, xε ) defined in (1.4), by
u1 x,
x
ε
= χj
x
∂u0
Qε
.
ε
∂xj
(1.12)
For u0 ∈ H 2 (Ω) we prove that
uε (·) − u0 (·) − εχj · Qε ∂u0 + εβε (·)
ε
∂x
j
H01 (Ω )
Cεu0 H 2 (Ω ) ,
(1.13)
106
D. Onofrei and B. Vernescu / Error estimates for periodic homogenization with non-smooth coefficients
where βε is defined by
−∇ · A
x
∇βε = 0
ε
in Ω,
βε = u1 x,
x
ε
on ∂Ω.
(1.14)
Cε2 u0 W 3,p (Ω ) .
(1.15)
Assuming u0 ∈ W 3,p (Ω) with p > N we obtain
uε (·) − u0 (·) − εχj · ∂u0 + εθε (·)
ε ∂x
L2 (Ω )
j
For completeness, in the Appendix, we give some of the definitions and properties related to the
unfolding operator (for which we refer to [7]), and we give the proofs for the convergence results related
to our smoothing argument.
As an application of (1.14) we mention that one can follow similar arguments as in [11] to prove the
convergence of the Multiscale Finite Element method proposed by T. Hou and X. Wu for the general
case of nonsmooth coefficients.
For a complete asymptotic analysis of the second order boundary layer corrector ϕε defined in (1.10),
together with new second order error estimates and their applications, we refer the reader to [17].
2. First order error estimates
The main result of this section is
Theorem 2.1. Let uε , u0 , u1 , and βε be defined as in Section 1. Then we have
uε (·) − u0 (·) − εu1 ·, . + εβε (·)
ε
H01 (Ω )
Cεu0 H 2 (Ω ) .
The proof of the theorem will be done in several steps:
Step 1. The first step is to consider the mollified coefficient matrix (Anij )N
i,j=1 , defined in the Appendix,
n
n
∞
∞
with the properties Aij L Aij L , (Aij ) is a Y -periodic matrix, and
Anij → Aij
in Lp (Y ) for 1 p < ∞.
(2.1)
For these coefficients the corresponding functions unε , χnj , un1 , and βεn defined similarly as in Section 1,
(1.1), (1.3), (1.12), and (1.14), respectively, satisfy (see Appendix):
χnj χj
1
in Hper
(Y ),
unε uε
n
in H01 (Ω),
un1 u1
in H 1 (Ω),
n
in H 1 (Ω).
βεn βε
(2.2)
Step 2. Next we define
v0n (x, y) = An (y)Qε (∇x u0 ) + An (y)∇y un1 (x, y)
(2.3)
D. Onofrei and B. Vernescu / Error estimates for periodic homogenization with non-smooth coefficients
107
or equivalently
n
v0 (x, y) i =
Anij (y) + Anik (y)
∂χnj
∂u0
Qε
.
∂yk
∂xj
(2.4)
By using the definition of χnj we have ∇y · v0n = 0. Let us denote by
∂χnj
n C (y) ij = Anij (y) + Anik (y)
∂y
k
n
and Ahom
n = MY (C (y)). It can be seen that
∇y · v0n − Ahom
n Qε (∇x u0 ) = 0.
(2.5)
Lemma 2.2. There exists q n (x, ·) ∈ [Wper (Y )]N such that curly q n = v0n − Ahom
n Qε (∇x u0 ).
Proof. Let B n (y) = C n (y) − Ahom
n . We then have
n
v0n − Ahom
n Qε (∇x u0 ) = B (y)Qε (∇x u0 ).
(2.6)
We look for q n of the form
q n (x, y) = φn (y)Qε (∇x u0 ),
where φn (y) = (φnij (y))ij with φnij (y) ∈ Wper (Y ).
If we denote by Bln the vector Bln = (Biln )i ∈ [L2per (Y )]N we observe that ∇y · Bln = 0. Hence
from the Theorem 3.4 in Girault and Raviart [9] adapted to the periodic case, the vectors φnl = (φnil )i ∈
[Wper (Y )]N are determined as the solutions to
curly φnl = Bln
divy φnl = 0.
and
(2.7)
Obviously we have
curly q n (x, y) = v0n − Ahom
n Qε (∇x u0 ).
(2.8)
Remark 2.3. From (2.2) it can be immediately seen that B n is bounded independently of n in
[L2 (Y )]N ×N and using the Appendix we have
Bn B
in L2 (Y )
N ×N
,
where B has an identical form as B n and it can be easily determined from the above limit. This together
with (2.7) and Theorem 3.9 in [9] adapted for the periodic case implies that φn is bounded independently
of n in (Wper (Y ))N ×N and we have
φnl φl
in Wper (Y )
curly φl = Bl
and
N
,
where
divy φl = 0;
for every l ∈ {1, . . . , N }.
(2.9)
108
D. Onofrei and B. Vernescu / Error estimates for periodic homogenization with non-smooth coefficients
Step 3. Next we define
v1n (x, y) = curlx q n (x, y)
and using Lemma 2.2 we have
∇y · v1n = −∇x · curly q n = −∇x · v0n − fεn ,
(2.10)
where
fεn = −∇x · Ahom
n Qε (∇x u0 ) .
We define
zεn (x) = unε (x) − u0 (x) − εun1 x,
µnε (x)
=A
x
,
ε
(2.11)
x
x
x
∇unε (x) − v0n x,
− εv1n x,
.
ε
ε
ε
n
(2.12)
From the above definitions, similarly as in [16] we obtain
A
n
x
x
x
x
∇zεn (x) − µnε (x) = ε v1n x,
− An
∇x un1 x,
ε
ε
ε
ε
+A
n
x Qε (∇x u0 ) − ∇x u0 .
ε
(2.13)
Next, we will prove that the L2 norm of (2.13) is of order ε. In order to do this we will show that v1n (x, xε )
and An ( xε )∇x u1 (x, xε ) are bounded in L2 independently of n and ε. We have the following estimate
Lemma 2.4. Let Ω ⊂ RN as before. For any ψ ∈ L2 (Y ), Y -periodic, we have
∇x Qε ∂u0 ψ x ∂x
ε j
L2 (Ω )
Cu0 H 2 (Ω ) ψL2 (Y ) .
Proof. We recall the definition of Qε
Qε (φ)(x) =
MYε (φ)(εξ
+
εi)x̄i1,1ξ
· ··· ·
x̄iNN,ξ ,
i1 ,...,iN
ξ=
x
ε
for any x ∈ Ω̃ε , with Ω̃ε defined in the Appendix, and any φ ∈ L2 (Ω̃ε,2 ) with Ω̃ε,2 = {x ∈
Ω; dist(x, Ω) < 2ε}, where i = (i1 , . . . , iN ) ∈ {0, 1}N and
x̄ikk,ξ =

xk − εξk



ε


 1 − xk − εξk
ε
if ik = 1,
if ik = 0,
x ∈ ε(ξ + Y ).
D. Onofrei and B. Vernescu / Error estimates for periodic homogenization with non-smooth coefficients
109
Note that, on each cube εξ + εY the first order derivative Qε takes the form
M ε (φ)(εξ + ε(1, i2 , . . . , in )) − M ε (φ)(εξ + ε(0, i2 , . . . , in ))
∂
Y
Y
Qε (φ)(x) =
∂x1
ε
i ,...,i
N
2
×
x̄i2,2ξ
· · · · · x̄iNN,ξ ,
where ξ = [ xε ]. Therefore we have
2 2
∂
ψ x dx
Q
(φ)(x)
∂x ε
ε εξ+εY
1
M ε (φ)(εξ + ε(1, i2 , . . . , in )) − M ε (φ)(εξ + ε(0, i2 , . . . , in )) 2
Y
Y
ε
2N −1
i2 ,...,iN
×
εξ+εY
2
ψ x dx
ε M ε (φ)(εξ + ε(1, i2 , . . . , in )) − M ε (φ)(εξ + ε(0, i2 , . . . , in )) 2
Y
Y
εN ψ2 2 .
L (Y )
ε
N −1
=2
i2 ,...,iN
(2.14)
Using the Schwartz inequality in (2.14) together with the definition of the mean MYε we obtain
εξ+εY
2 2
∂
ψ x dx
Q
(φ)(x)
∂x ε
ε 1
CψL2 (Y )
2
i2 ,...,iN
CψL2 (Y )
2
εξ+εY
φ(x + ε(1, i2 , . . . , in )) − φ(x + ε(0, i2 , . . . , in )) 2
dx
ε
εξ+εY
φ(x + ε(1, i2 , . . . , in )) − φ(x) 2
ε
i2 ,...,iN
φ(x + ε(0, i2 , . . . , in )) − φ(x) 2
dx.
+
ε
After summing the above inequalities over ξ ∈ {ξ ∈ ZN ; (εξ +εY )∩Ω = ∅}, and using the inequality
between the differential quotients and the gradient we obtain
2 2
∂
ψ x Cψ2 2 ∇φ2 2
Q
(φ)(x)
.
∂x ε
L (Y )
L (Ω̃ε,2 )
ε Ω
1
This yields
Ω
2
∇x Qε (φ)2 ψ x Cψ2 2 ∇φ2 2
.
L (Y )
L (Ω̃ε,2 )
ε 110
D. Onofrei and B. Vernescu / Error estimates for periodic homogenization with non-smooth coefficients
Recall that, u0 denotes the stable extension of u0 to the whole space. Therefore, choosing φ to be the
partial derivative of u0 the conclusion of the lemma follows.
Proof of Theorem 2.1. By applying Lemma 2.4 we can see that
n x
x n
A
∇x u1 x,
ε
ε L2 (Ω )
C χnj 2L2 (Y ) u0 H 2 (Ω ) Cu0 H 2 (Ω )
(2.15)
and using Remark 2.3 we obtain
n
v x, x 1
ε L2 (Ω )
12
n 2
C
φij L2 (Y ) u0 H 2 (Ω ) Cu0 H 2 (Ω ) .
(2.16)
i,j
Using (2.15), (2.16) and the properties of Qε we obtain the following estimate for the left hand side
of (2.13):
n x
n
n
A
∇zε (x) − µε (x)
ε
L2 (Ω )
Cεu0 H 2 (Ω ) .
(2.17)
For g ∈ L2 (Ω) we define wεn ∈ H01 (Ω) solution of the following problem
x
∇wεn = g
ε
−∇ · An
wεn = 0 on ∂Ω.
in Ω,
(2.18)
Obviously we have
n
w ε H01 (Ω )
gH −1 (Ω ) .
(2.19)
Using zεn + εβεn as a test function in (2.18), with βε defined by (1.14) we obtain
Ω
n
zε + εβεn g dx =
A
n
Ω
x
∇zεn · ∇wεn dx.
ε
(2.20)
The right hand side can be estimated as follows
An
Ω
x
∇zεn · ∇wεn dx =
ε
An
Ω
x
∇zεn − µnε · ∇wεn dx −
ε
n x
n
n
A
∇z
−
µ
ε
ε
ε
n
w L2 (Ω )
ε H01 (Ω )
Ω
∇ · µnε wεn dx
+ ∇ · µnε H −1 (Ω ) wεn H 1 (Ω ) .
0
(2.21)
We note here that ∇ · µnε ∈ L2 (Ω); indeed:
∇ · µnε (x) = ∇ · An
x
x
1
x
∇unε (x) − ∇x · v0n x,
− ∇y · v0n x,
ε
ε
ε
ε
D. Onofrei and B. Vernescu / Error estimates for periodic homogenization with non-smooth coefficients
− ∇x · v1n x,
x
x
− ∇y · v1n x,
ε
ε
111
= −f (x) − ∇x · Ahom
n Qε (∇x u0 ) .
To estimate the H −1 norm of ∇ · µnε we consider φ ∈ H01 (Ω) and
Ω
∇·
µnε φ(x) dx =
(A
Ω
hom
Qε (∇u0 ) − A
hom
∇u0 )∇φ dx +
C ∇u0 − Qε (∇u0 )
Ω
hom
Ahom
Qε (∇u0 )∇φ dx
n −A
L2 (Ω ) φH01 (Ω )
+ φH 1 (Ω )
0
hom
A
− Ahom Qε (∇u0 )
n
L2 (Ω )
Cεu0 H 2 (Ω ) φH 1 (Ω ) + Kn φH 1 (Ω ) u0 H 1 (Ω ) ,
0
0
(2.22)
.
where we used the properties of Qε and Kn = |Ahom − Ahom
n |.
Therefore we proved that
∇ · µn ε H −1 (Ω )
Cεu0 H 2 (Ω ) + Kn u0 H 1 (Ω ) .
(2.23)
Thus (2.17) and (2.23) used in (2.22) imply
n
n
n
Cεu0 2 wn 1
z
+
εβ
g
dx
H (Ω )
ε
ε
ε H0 (Ω ) + CKn wε H01 (Ω )
Ω
Cεu0 H 2 (Ω ) gH −1 (Ω ) + CKn gH −1 (Ω ) ,
where we used (2.19). From the above inequality we have
n
z + εβ n ε
ε H01 (Ω )
Cεu0 H 2 (Ω ) + CKn .
(2.24)
From (2.1) and (2.2) we have that Kn → 0 as n → ∞. Using the Appendix we can pass to the limit
when n → ∞ in (2.24) and from (2.2) we get
zε + εβε H 1 (Ω ) Cεu0 H 2 (Ω )
0
which is exactly what needs to be proved.
3. The error estimate in the L2 norm
In this section we will look for minimal assumptions on u0 needed to prove the classical error estimates
(1.9) in the case of non-smooth coefficients.
The L2 -norm of
uε (·) − u0 (·) − εw1 ·,
.
ε
+ εθε (·)
(3.1)
112
D. Onofrei and B. Vernescu / Error estimates for periodic homogenization with non-smooth coefficients
with w1 (·, ε. ) as defined in (1.4), can be estimated with additional assumptions. In [16], Moscow and
Vogelius proved the estimate (1.9) assuming that u0 ∈ H 3 (Ω) and A ∈ C ∞ (Y ). In [21], Sarkis and
1,q
Versieux prove an estimate of order ε2 , under the assumptions that u0 ∈ W 3,p (Ω) and χj , χij ∈ Wper
(Y )
1
1
1
for p, q > N where p + q 2 .
We extend in this section the previous results, by only requiring that u0 ∈ W 3,p (Ω) for p > N , to
prove (1.9) in the case of nonsmooth coefficients. In order to do this we need to introduce the second
order cell problems. Therefore, let χnij ∈ Wper (Y ) denote the solutions of
∇y · An ∇y χnij = bnij − MY bnij ,
(3.2)
where
bnij = −Anij − Anik
∂χnj
∂ n n
−
A χ
∂yk
∂yk ik j
and MY (·) is the average on Y . From Corollary A.9, in the Appendix
∇y χn ij L2 (Y )
<C
and
χnij χij
∀i, j ∈ {1, . . . , N },
in Wper (Y ),
where
Y
A(y)∇y χij ∇y ψ dy = bij − MY (bij ), ψ
(Wper (Y ),(Wper (Y )) )
for any ψ ∈ Wper (Y ) and with
bij = −Aij − Aik
∂χj
∂
−
(Aik χj ).
∂yk
∂yk
Next, we will only assume that u0 ∈ W 3,p (Ω) with N < p ∞ to prove the estimate of order ε2 for
(3.1). Indeed we have,
Theorem 3.1. Let uε , u0 , u1 and θε defined as in Section 2. If u0 ∈ W 3,p (Ω), N < p ∞ we have
uε (·) − u0 (·) − εw1 ·, . + εθε (·)
ε
L2 (Ω )
Cε2 u0 W 3,p (Ω ) .
(3.3)
Proof. For the sake of simplicity we will consider only the case when N = 3, the two dimensional case
being similar. As in the previous section we can assume the smooth coefficients An (see (A.2)), and
follow the same ideas as in [16] to define
un2 (x, y) = χnij (y)
∂2 u 0
(x).
∂xj ∂xi
For p > N we have that
∇x un ·, . 2
ε L2 (Ω )
n . χij ε 2p
L p−2 (Ω )
2
∇x ∂ u0
∂x ∂x
j
i Lp (Ω )
D. Onofrei and B. Vernescu / Error estimates for periodic homogenization with non-smooth coefficients
113
and using a change in variables and the inequality (A.11) in the Appendix, we obtain
2
∇x un ·, . 2
ε 2
C
L (Ω )
2
∇x ∂ u0
∂x ∂x
j
i,j
2
p
i L (Ω )
Cu0 2W 3,p (Ω ) .
(3.4)
As in [16] we will define
∂χnij ∂2 u0
n
∂2 u 0
v∗ (x, y) k = Anki (y)χnj (y)
(x) + Ankl (y)
.
∂xj ∂xi
∂yl ∂xj ∂xi
(3.5)
Following similar arguments we can observe that ∇x · MY (v∗n ) = 0. By introducing
j
Rki
= MY Anki χnj + Ankl
∂χnij
.
∂yl
n ∈ [L2 (Y )]3 defined by,
Consider αij




∂χn
ij
∂y l
∂χn
An2l ∂yijl
∂χn
An3l ∂yijl
An1i χnj + An1l
n
αij
=  An2i χnj +
An3i χnj +
− R1ji



n
+ βij
− R2ji 

− R3ji
with
T
T
T
β1nj = 0, −φn3j , φn2j ,
β2nj = φn3j , 0, −φn1j ,
for j ∈ {1, 2, 3},
β3nj = −φn2j , φn1j , 0 ,
where T denotes the transpose and φnij are defined at (2.7). Using the symmetry of the matrix A we
n defined above are divergence free with zero average over Y . This implies
observe that the vectors αij
n
that there exists ψij ∈ [Wper (Y )]3 , (see Theorem 3.4, [9] adapted to the periodic case) so that
n
n
curly ψij
= αij
and
n
∇ · ψij
= 0 for any i, j ∈ {1, 2, 3}.
(3.6)
From Corollaries A.5 and A.9 in Appendix and we observe that
n
αij
αij
3
in L2 (Y ) ,
(3.7)
n and can be obviously obtain from (3.7). Using the
where the form of αij is identical with that of αij
above convergence result and Theorem 3.9 from [9] adapted to the periodic case, we obtain that
n
ψij ,
ψij
in Wper (Y ) for any i, j ∈ {1, 2, 3}
114
D. Onofrei and B. Vernescu / Error estimates for periodic homogenization with non-smooth coefficients
and ψij satisfy
curly ψij = αij
and
∇y · ψij = 0 for i, j ∈ {1, 2, 3}.
(3.8)
2
n (y) ∂ u0 (x) and v n (x, y) = curl pn (x, y). Obviously we have that
Next define pn (x, y) = ψij
x
2
∂x j ∂x i
∇x · v2n = 0. It is also easy to check, that ∇y · v2n = −∇x · v∗n , (see [16] for example). We set
w1n (x, y) = χnj (y)
∂u0
(x),
∂xj
r0n (x, y) = An (y)∇x u0 + An (y)∇y w1n (x, y),
ψεn (x)
=
unε (x)
− u0 (x) −
εw1n
ξεn (x)
=A
n
x
x
x,
− ε2 un2 x,
,
ε
ε
(3.9)
x
x
x
x
∇unε − r0n x,
− εv∗n x,
− ε2 v2n x,
.
ε
ε
ε
ε
(3.10)
As in [16] we can write
A
n
x
x
x
x
∇ψεn (x) − ξεn (x) = ε2 v2n x,
− An
∇x un2 x,
ε
ε
ε
ε
.
(3.11)
We use next, as in (3.4), the inequality (A.11) to obtain
n
v ·, . 2
ε L2 (Ω )
Cu0 W 3,p (Ω ) .
(3.12)
Using (3.4), (3.11) and (3.12) we get
n x
n
n
A
∇ψε (x) − ξε (x)
ε
L2 (Ω )
Cε2 u0 W 3,p (Ω ) .
(3.13)
Similarly as in [16] we have that ∇ · ξεn (x) = 0. Let us define ϕnε as solution of
∇· A
n
x
∇ϕnε = 0 in Ω,
ε
ϕnε
=
un2
x
x,
ε
on ∂Ω.
(3.14)
Using again Corollary A.11 in Appendix, we have that ϕnε ϕε in H 1 (Ω) where ϕε is the solution
of
∇· A
x
∇ϕε = 0 in Ω,
ε
ϕε = u2 x,
x
ε
on ∂Ω.
(3.15)
Then,
. ϕε L2 (Ω ) C u2 ·,
ε L∞ (∂Ω )
Cχij L∞ (Y ) u0 W 3,p (Ω ) Cu0 W 3,p (Ω ) ,
(3.16)
D. Onofrei and B. Vernescu / Error estimates for periodic homogenization with non-smooth coefficients
115
where we used [13] for the L∞ bound on χij . Next, similarly as in [16] we have
n
u (·) − u0 (·) − εwn ·, . + εθ n (·) − ε2 un ·, . + ε2 ϕn ε
ε
1
2
ε
ε
ε
L2 (Ω )
Cε2 u0 W 3,p (Ω )
and passing to the limit when n → ∞ using triangle inequality, (3.4) and (3.16) we get (3.3).
Remark that the assumption that u0 ∈ W 3,p , with p > N was necessary for the estimate (3.16).
Appendix A
In this section we will present the proofs for some of the results used in the previous sections and
which were not included in the main body of the chapter for the sake of clarity of the exposition.
A.1. Definition and properties of the unfolding operator
Let Ξε = {ξ ∈ ZN ; (εξ + εY ) ∩ Ω = ∅} and define
Ω̃ε =
(εξ + εY ).
(A.1)
ξ∈Ξε
1
∞ (Y ) in the H 1 norm, where C ∞ (Y ) is the subset
Let us also consider Hper
(Y ) to be the closure of Cper
per
∞
N
of C (R ) of Y -periodic functions, and
1
.
1
Wper (Y ) = v ∈ Hper
(Y )/R,
|Y |
v dy = 0
Y
(see [6] for properties).
Next we present several very useful technical inequalities obtained in [10].
Proposition A.1. We have:
(1)
(2)
(3)
(4)
ψ . ε L2 (Ω̂ε )
. + ∇y ψ
ε L2 (Ω )
vL2 (Ω̃ε )
ε
M (v)
Y
L2 (Ω̂ε )
1
Cε 2 ψH 1 (Y )
for any v ∈ L2 (Ω̃ε ).

v − M ε (v)
2

Y

L (Ω ) Cε∇v[L2 (Ω )]N ,


v − Tε (v)
2
L (Ω×Y ) Cε∇v[L2 (Ω )]N ,



 Q (v) − M ε (v)
2
Cε∇v 2
ε
Qε (v)ψ . ε Y
L2 (Ω )
L (Ω )
1
for every ψ ∈ Hper
(Y ).
[L (Ω )]N
CvL2 (Ω̃ε,2 ) ψL2 (Y )
for any v ∈ H 1 (Ω).
for any v ∈ L2 (Ω̃ε,2 ) and ψ ∈ L2 (Y ).
116
D. Onofrei and B. Vernescu / Error estimates for periodic homogenization with non-smooth coefficients
A.2. Convergence results and the smoothing argument
Let mn ∈ C ∞ be the standard mollifying sequence, i.e., 0 < mn 1, RN mn dz = 1, sppt(mn ) ⊂
B(0, n1 ). Define An (y) = (mn ∗ A)(y), where a has been defined in the Introduction (see (1.1)). We
have:
(1) An − Y -periodic matrix,
(2) |An |L∞ < |A|L∞ ,
(3) An → A in Lp for any p ∈ (1, ∞).
(A.2)
From (A.2) we have that c|ξ|2 Anij (y)ξi ξj C|ξ|2 ∀ξ ∈ RN . Define
An
hom
ij
= MY
where MY (·) =
1
|Y | Y
Anij (y)
+
Anik (y)
∂χnj
,
∂yk
(A.3)
· dy and χnj ∈ Wper (Y ) are the solutions of the local problem
−∇y · A(y) ∇χnj + ej
= 0.
(A.4)
Next we present a few important convergence results needed in the smoothing argument developed in
the previous sections.
Lemma A.2. Let fn , f ∈ H −1 (Ω) with fn f in H −1 (Ω) and let bn , b ∈ L∞ (Ω), with
c|ξ|2 bnij (y)ξi ξj C|ξ|2 ,
c|ξ|2 bij (y)ξi ξj C|ξ|2
for all ξ ∈ RN and
bn → b
in L2 (Ω).
Consider ζn ∈ H01 (Ω) the solution of
b (x)∇ζn ∇ψ dx =
n
Ω
Ω
fn ψ dx
for any ψ ∈ H01 (Ω). Then we have
ζn ζ
in H01 (Ω)
and ζ verifies
b(x)∇ζ∇ψ dx =
Ω
Ω
f ψ dx for any ψ ∈ H01 (Ω).
D. Onofrei and B. Vernescu / Error estimates for periodic homogenization with non-smooth coefficients
117
Proof. Immediately can be observed that
ζn H 1 (Ω ) C
0
and therefore there exists ζ such that for a subsequence still denoted by n we have
in H01 (Ω).
ζn ζ
(A.5)
For any smooth ψ ∈ H01 (Ω) easily it can be seen that
Ω
bn (x)∇ζn ∇ψ dx →
b(x)∇ζ∇ψ dx
Ω
and this implies the statement of the lemma. Due to the uniqueness of ϕ one can see that the limit (A.5)
holds on the entire sequence. Remark A.3. Using similar arguments it can be proved that the results of Lemma A.2 hold true if we
replace the Dirichlet boundary conditions with periodic boundary conditions.
Corollary A.4. Let unε ∈ H01 (Ω) be the solution of


 −∇ · An x ∇un = f
ε
ε

 n
in Ω,
uε = 0
on ∂Ω.
We then have
n
unε uε
in H01 (Ω),
where uε verifies


x
−∇ · A
∇uε = f
ε

uε = 0
in Ω,
on ∂Ω.
Proof. Using (A.2) we have that
An
x
ε
n
→A
x
ε
in L2 (Ω)
and the statement follows immediately from Remark A.3. Corollary A.5. For j ∈ {1, . . . , N }, let χnj ∈ Wper (Y ) be the solution of
−∇y · An (y) ∇χnj + ej
= 0,
(A.6)
118
D. Onofrei and B. Vernescu / Error estimates for periodic homogenization with non-smooth coefficients
where {ej }j denotes the canonical basis of RN . Then we have
χnj χj
in Wper (Y ),
where χj ∈ Wper (Y ) verifies
−∇y · A(y)(∇χj + ej ) = 0.
Proof. From (A.2) we obtain
∂
∂ n
Aij (y) Aij (y)
∂yi
∂yi
in Wper (Y ) .
The statement of the remark follows then immediately from Remark A.3.
Proposition A.6. Let v ∈ [H 1 (Ω)]N be arbitrarily fixed and for every j ∈ {1, . . . , N }, let χj ∈
Wper (Y ) be defined as in (A.4), and χnj ∈ Wper (Y ), for j ∈ {1, . . . , N }, to be the solutions of (A.6).
Define
hnε (x) = hn x,
gεn (x) = g n x,
x
ε
x
ε
x
vj ,
ε
= χnj
hε (x) = h x,
= χj
x
Qε (vj ),
ε
= χnj
x
ε
x
ε
gε (x) = g x,
x
vj ,
ε
= χj
x
Qε (vj ).
ε
We have that
n
(1) gεn gε in H 1 (Ω).
(2) If v ∈ [W 1,p (Ω)]N , p > N ,
n
then hnε hε
in H 1 (Ω).
Proof. First note that applying Corollary A.5 to the sequence {χnj }n we have
n
χnj χj
in Wper (Y ).
(A.7)
Next we have
2
n
g x, x ε =
Ω
H 1 (Ω )
x
Qε (vj )
ε
χnj
+
Ω
2
χnj
1
ε2
dx +
x
∇x Qε (vj )
ε
Ω
∇y χnj
x
Qε (vj )
ε
2
dx
2
dx
(A.8)
and
2
n
h x, x ε H 1 (Ω )
=
Ω
2
x
vj
ε
χnj
+
Ω
χnj
dx +
x
∇x vj
ε
1
ε2
Ω
∇y χnj
2
x
vj
ε
dx
2
dx.
(A.9)
D. Onofrei and B. Vernescu / Error estimates for periodic homogenization with non-smooth coefficients
119
For the first convergence in Theorem A.6 we use that
n x
χ
j ε Qε (vj )
H 1 (Ω )
C χnj Wper (Y ) .
(A.10)
Next we can see that (A.8) imply that
2
n
g x, x − g x, x ε
ε 2
=
Ω
L (Ω )
χnj
2
x
x
− χj
ε
ε
2
Qε (vj ) dx
and using (A.10) we obtain the desired result.
For the second convergence result in Proposition A.6 we will recall now a very important inequality
(see [13], Chp. 2) to be used for our estimates.
For any p > N we have
φ
2p
L p−2 (Ω )
1− N
N
c(p) φL2 (Ω ) + ∇φLp2 (Ω ) φL2 (Ωp )
(A.11)
for any φ ∈ H 1 (Ω) and where c(p) is a constant which depends only on q, N , Ω.
For v ∈ [W 1,p (Ω)]N with p > N , using (A.7), the Sobolev embedding W 1,p (Ω) ⊂ L∞ (Ω) and (A.11)
in (A.9) we obtain
2
n
h x, x ε < C,
H 1 (Ω )
where the constant C above does not depend on n.
Next we can easily observe that
2
n
h x, x − h x, x ε
ε 2
L (Ω )
=
Ω
χnj
2
x
x
− χj
ε
ε
(vj )2 dx
and in either of the above cases, (A.7) and a few simple manipulations imply that
hn x,
x
ε
n
→ h x,
x
ε
in L2 (Ω).
This together with the bound on the sequence {hn (x, xε )}n implies the statement of the proposition.
The two convergence results in the next corollary will follow immediately from Proposition A.5.
Corollary A.7. Let w1nε (x) = w1n (x, xε ) = χnj ( xε ) ∂∂xuj0 and un1ε = un1 (x, xε ) = χnj ( xε )Qε ( ∂∂xuj0 ). Then we
have
(1) If u0 ∈ W 3,p (Ω) for p > N ,
n
w1nε w1
in H 1 (Ω).
120
D. Onofrei and B. Vernescu / Error estimates for periodic homogenization with non-smooth coefficients
(2) If u0 ∈ H 2 (Ω),
n
un1ε u1
in H 1 (Ω),
where w1 and u1 were defined in (1.4) and (1.12) respectively.
Corollary A.8. Let θεn be the solution of
x
∇θεn = 0 in Ω,
ε
−∇ · An
θεn = w1n x,
x
ε
on ∂Ω
(A.12)
on ∂Ω.
(A.13)
and βεn be the solution of
−∇ · A
x
∇βεn = 0 in Ω,
ε
n
βεn
=
un1
x
x,
ε
We have that
(i) if u0 ∈ W 3,p (Ω), p > N , then
n
θεn θε
in H 1 (Ω).
(ii) if u0 ∈ H 2 (Ω), then
n
βεn βε
in H 1 (Ω),
where θε and βε satisfies
x
−∇ · A
∇θε = 0 in Ω,
ε
x
x,
ε
θ ε = w1
on ∂Ω
(A.14)
on ∂Ω.
(A.15)
and
x
−∇ · A
∇βε = 0
ε
in Ω,
βε = u1
x
x,
ε
Proof. Using Corollary A.7 and a few simple arguments one can simply show that
−∇ · A
x
x
∇w1n x,
ε
ε
n
−∇ · A
x
x
∇w1 x,
ε
ε
in H −1 (Ω)
and
x
x
∇w1n x,
−∇ · A
ε
ε
x
x
−∇ · A
∇w1 x,
ε
ε
n
in H −1 (Ω).
Homogenizing the data in the problems (A.12) and (A.13) and using Corollary A.7 and Lemma A.2
the statement follows immediately. D. Onofrei and B. Vernescu / Error estimates for periodic homogenization with non-smooth coefficients
121
Corollary A.9. For any i, j ∈ {1, . . . , N } let χnij ∈ Wper (Y ) be the solutions of :
∇y · An ∇y χnij = bnij − MY bnij ,
(A.16)
where
bnij = −Anij − Anik
∂χnj
∂ n n
−
A χ
∂yk
∂yk ik j
and MY (·) is the average on Y .
Then we have
χnij χij
in Wper (Y ) for any i, j ∈ {1, . . . , N },
where χij satisfies
Y
A(y)∇y χij ∇y ψ dy = bij − MY (bij ), ψ
((Wper (Y )) ,Wper (Y ))
(A.17)
for any ψ ∈ Wper (Y ) and with
bij = −Aij − Aik
∂χj
∂
−
(Aik χj ).
∂yk
∂yk
Proof. For any ψ ∈ Wper (Y ), we have that,
Y
n
bij − MY bnij ψ dy =
Y
−Anij
−
Anik
∂χnj
hom ∂ψ
ψ dy + An ij
ψ dy +
Anki χnj
dy,
∂yk
∂yk
Y
Y
(A.18)
where we used MY (bnij ) = −(Ahom
n )ij (see [16]).
Using (A.2), (A.7), and simple manipulations we can prove that
Anik
∂χnj
∂χj
Aik
∂yk
∂yk
in L2 (Y )
(A.19)
and
Anik χnj Aik χj
in L2 (Y ).
(A.20)
From (A.19), (A.2) and (A.3) we have that
Ahom
n
ij
→ Ahom
ij .
(A.21)
Finally using (A.2), (A.7), (A.19) and (A.20) in (A.18) we obtain that
bnij − MY bnij bij − MY (bij )
in Wper (Y ) .
This and Remark A.3 complete the proof of the statement.
122
D. Onofrei and B. Vernescu / Error estimates for periodic homogenization with non-smooth coefficients
Remark A.10. We can easily observe that we have
n
Anij χnij Aij χij ,
Anij
∂χnij n
∂χij
Aij
∂yk
∂yk
weakly in Wper (Y ).
Corollary A.11. Let u0 ∈ H 2 (Ω) be the solution of the homogenized problem (1.2) and χnij , χij ∈
Wper (Y ) be defined by (A.16) and (A.17). Suppose that there exists p > N such that u0 ∈ W 3,p (Ω).
2
2
Define un2 (x, y) = χnij (y) ∂∂xj u∂0xi (x) and u2 (x, y) = χij (y) ∂∂xj u∂0xi (x). Consider ϕnε the solution of
∇· A
x
∇ϕnε = 0 in Ω,
ε
n
ϕnε
=
un2
x
x,
ε
on ∂Ω.
(A.22)
Then we have that
un2
x
x,
ε
n
u2
x
x,
ε
and
n
ϕnε ϕε
in H 1 (Ω),
where ϕε satisfies
x
∇· A
∇ϕε = 0 in Ω,
ε
ϕε = u2
x
x,
ε
on ∂Ω.
(A.23)
Proof. Following similar arguments as those used in Corollary A.6 we can prove that
un2
x
x,
ε
n
χij (y)
∂2 u 0
(x) in H 1 (Ω).
∂xj ∂xi
Using the above convergence result and similar ideas as in Corollary A.8 we complete the proof of the
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